1. Section 1.1 Introduction to Graphing Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

2. Objectives  Plot points.  Determine whether an ordered pair is a solution of an equation.  Find the x-and y-intercepts of an equation of the form Ax + By = C.  Graph equations.  Find the distance between two points in the plane and find the midpoint of a segment.  Find an equation of a circle with a given center and radius, and given an equation of a circle in standard form, find the center and the radius.  Graph equations of circles.

3. Cartesian Coordinate System

4. Example To graph or plot a point, the first coordinate tells us to move left or right from the origin. The second coordinate tells us to move up or down. Plot (  3, 5). Move 3 units left. Next, we move 5 units up. Plot the point. (–3, 5)

5. Solutions of Equations Equations in two variables have solutions (x, y) that are ordered pairs. Example: 2x + 3y = 18 When an ordered pair is substituted into the equation, the result is a true equation. The ordered pair has to be a solution of the equation to receive a true statement.

6. Examples a. Determine whether the ordered pair (  5, 7) is a solution of 2x + 3y = 18. 2(  5) + 3(7) ? 18  10 + 21 ? 18 11 = 18 FALSE (  5, 7) is not a solution. b. Determine whether the ordered pair (3, 4) is a solution of 2x + 3y = 18. 2(3) + 3(4) ? 18 6 + 12 ? 18 18 = 18 TRUE (3, 4) is a solution.

7. Graphs of Equations To graph an equation is to make a drawing that represents the solutions of that equation.

8. x-Intercept The point at which the graph crosses the x-axis. An x-intercept is a point (a, 0). To find a, let y = 0 and solve for x.

9. Example Find the x-intercept of 2x + 3y = 18. 2x + 3(0) = 18 2x = 18 x = 9 The x-intercept is (9, 0).

10. y-Intercept The point at which the graph crosses the y-axis. A y-intercept is a point (0, b). To find b, let x = 0 and solve for y.

11. Example Find the y-intercept of 2x + 3y = 18. 2(0) + 3y = 18 3y = 18 y = 6 The y-intercept is (0, 6).

12. Example Graph 2x + 3y = 18. We already found the x-intercept: (9, 0) We already found the y-intercept: (0, 6) We find a third solution as a check. If x is replaced with 5, then Thus, is a solution.

13. Example (continued) Graph: 2x + 3y = 18. x-intercept: (9, 0) y-intercept: (0, 6) Third point:

14. Example Graph y = x 2 – 9x – 12. (12, 24)2412 –2  32  26  12 –2 24 y (10, –2)10 (5,  32) 5 (4,  32) 4 (2,  26) 2 (0,  12) 0 (  1, –2) 11 (  3, 24) 33 (x, y)x Make a table of values.

15. The Distance Formula The distance d between any two points (x 1, y 1 ) and (x 2, y 2 ) is given by

16. Example Find the distance between the points (–2, 2) and (3,  6).

17. Midpoint Formula If the endpoints of a segment are (x 1, y 1 ) and (x 2, y 2 ), then the coordinates of the midpoint are

18. Example Find the midpoint of a segment whose endpoints are (  4,  2) and (2, 5).

19. Circles A circle is the set of all points in a plane that are a fixed distance r from a center (h, k). The equation of a circle with center (h, k) and radius r, in standard form, is (x  h) 2 + (y  k) 2 = r 2.

20. Example Find an equation of a circle having radius 5 and center (3,  7). Using the standard form, we have (x  h) 2 + (y  k) 2 = r 2 [x  3] 2 + [y  (  7)] 2 = 5 2 (x  3) 2 + (y + 7) 2 = 25.